### Abstract

For fixed integers p, q an edge coloring of a complete graph K is called a (p, q)-coloring if the edges of every K_{p} ⊆ K are colored with at least q distinct colors. Clearly, (p, 2)-colorings are the classical Ramsey colorings without monochromatic K_{p} subgraphs. Let f(n,p,q) be the minimum number of colors needed for a (p,q)-coloring of K_{n}. We use the Local Lemma to give a general upper bound for f. We determine for every p the smallest q for which f(n,p,q) is linear in n and the smallest q for which f(n,p,q) is quadratic in n. We show that certain special cases of the problem closely relate to Turán type hypergraph problems introduced by Brown, Erdos and T. Sós. Other cases lead to problems concerning proper edge colorings of complete graphs.

Original language | English |
---|---|

Pages (from-to) | 459-467 |

Number of pages | 9 |

Journal | Combinatorica |

Volume | 17 |

Issue number | 4 |

Publication status | Published - 1997 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*17*(4), 459-467.

**A variant of the classical Ramsey problem.** / Erdős, P.; Gyárfás, A.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 17, no. 4, pp. 459-467.

}

TY - JOUR

T1 - A variant of the classical Ramsey problem

AU - Erdős, P.

AU - Gyárfás, A.

PY - 1997

Y1 - 1997

N2 - For fixed integers p, q an edge coloring of a complete graph K is called a (p, q)-coloring if the edges of every Kp ⊆ K are colored with at least q distinct colors. Clearly, (p, 2)-colorings are the classical Ramsey colorings without monochromatic Kp subgraphs. Let f(n,p,q) be the minimum number of colors needed for a (p,q)-coloring of Kn. We use the Local Lemma to give a general upper bound for f. We determine for every p the smallest q for which f(n,p,q) is linear in n and the smallest q for which f(n,p,q) is quadratic in n. We show that certain special cases of the problem closely relate to Turán type hypergraph problems introduced by Brown, Erdos and T. Sós. Other cases lead to problems concerning proper edge colorings of complete graphs.

AB - For fixed integers p, q an edge coloring of a complete graph K is called a (p, q)-coloring if the edges of every Kp ⊆ K are colored with at least q distinct colors. Clearly, (p, 2)-colorings are the classical Ramsey colorings without monochromatic Kp subgraphs. Let f(n,p,q) be the minimum number of colors needed for a (p,q)-coloring of Kn. We use the Local Lemma to give a general upper bound for f. We determine for every p the smallest q for which f(n,p,q) is linear in n and the smallest q for which f(n,p,q) is quadratic in n. We show that certain special cases of the problem closely relate to Turán type hypergraph problems introduced by Brown, Erdos and T. Sós. Other cases lead to problems concerning proper edge colorings of complete graphs.

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UR - http://www.scopus.com/inward/citedby.url?scp=0013334029&partnerID=8YFLogxK

M3 - Article

VL - 17

SP - 459

EP - 467

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -